From the definition of the Cartesian product we have that an arbitrary object $a$ is in $X_1 \times X_2 \times ... \times X_n$ if and only if for every component $a_i$ of $a$ there is an element $x_i$ in $X_i$ such that $a_i = x_i$.

$\leftarrow$: If at least one $X_i$ is empty there is no such $x_i$.

$\rightarrow$: If $X_1 \times X_2 \times ... \times X_n$ is empty, there is no tuple $a$. $\Box$

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